Existence Of Global Solution And Blow-up Solution To Several Classes Of Nonlinear Systems

Nonlinear parabolic system is an important aspect of partial diferential equa-tion theory, so the studying of the qualitative of the solution is a very importantdirection. In this thesis, three classes of nonlinear parabolic systems will be stud-ied. The existence of global solutions, blow-up in fnite time and some blow upproperties will be studied, especially the blow-up rate estimates. This thesis iscomposed of four chapters.In chapter1, the background and related results of these nonlinear systemswill be presented, and the main work of this thesis will also be introduced.Chapter2deals with the related properties of solutions of the nonlinearparabolic system with nonlocal sources and nonlocal boundaries. We will show howthe weighted functions in the boundary, nonlocal sources and difusion afect theproperties of solutions. By constructing auxiliary functions and comparison prin-ciple and upper-lower solution arguments, we obtain the conditions on the globalexistence and fnite time blow-up solutions. Also, the blow-up rate estimates willbe established.In chapter3, we concern a class of nonlinear parabolic system with absorptionsunder nonlocal boundaries. We will show how the nonlocal sources, absorptionsand weighted functions in the boundary conditions afect the properties of solu-tions. By constructing auxiliary functions and combining with related theorem,inequality, the condition on the global existence and blow-up in fnite time of so-lutions will be given out. Furthermore, using some technique, the blow-up rateestimates of the blow-up solutions will also be established.In chapter4, we consider the blow-up properties of solutions to a nonlinearparabolic system with weighted local terms and Dirichlet boundary conditions.The main aim of this chapter is to study the infuence of the two sources: localsources un(0, t) and vq(0, t), and the weighted functions a(x) and b(x) on theasymptotic behaviors of solutions. Firstly, by using upper and lower solutions,the conditions on the global existence and blow-up in fnite of solutions will begiven out. Then, by contradiction, single point blow-up will be proved under someconditions. Moreover, the blow-up rate estimates in the case of m, p≤1will alsobe obtained.